Chapter 4.1, Problem 18E

a.

To determine

Find the complement and supplement for the given angle.

Answer to Problem 18E

The complement of the angle is $\frac{5\pi }{6}$ and the supplementis $\frac{11\pi }{12}$ .

Explanation of Solution

Given:

The given angle is $\frac{\pi }{12}$ .

Calculation:

Two positive angles $\alpha$ and $\beta$ are complementary (complements of each other) when their sum is $\frac{\pi }{2}$ . Two positive angles $\alpha$ and $\beta$ are supplementary (supplement of each other) when their sum is $\pi$ .

The complement of $\frac{\pi }{12}$ is

  $\frac{\pi }{2}-\frac{\pi }{12}=\frac{5\pi }{6}$

The supplement of $\frac{\pi }{12}$ is

  $\pi -\frac{\pi }{12}=\frac{11\pi }{12}$

Hence the complement of the angle is $\frac{5\pi }{6}$ and the supplementis $\frac{11\pi }{12}$ .

b.

To determine

Find the complement and supplement for the given angle.

Answer to Problem 18E

The supplement is $\frac{\pi }{12}$ and there is no complement.

Explanation of Solution

Given:

The given angle is $\frac{11\pi }{12}$ .

Calculation:

Two positive angles $\alpha$ and $\beta$ are complementary (complements of each other) when their sum is $\frac{\pi }{2}$ . Two positive angles $\alpha$ and $\beta$ are supplementary (supplement of each other) when their sum is $\pi$ .

The supplement of $\frac{11\pi }{12}$ is

  $\pi -\frac{11\pi }{12}=\frac{\pi }{12}$

The given angle $\frac{11\pi }{12}$ is greater than $\frac{\pi }{2}$ , there is no complement.

Hence the supplement is $\frac{\pi }{12}$ andthere is no complement.